Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds

Abstract

We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension $n\ge 3$, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant $δ_0$, the flat set has no $C^1$ hypersurface piece; it is empty when $δ_0>1/2$, and is locally porous when $δ_0=1/2$. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly $1/q$-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is \[ δ_c=\frac1{n-1}. \] A locally uniform gap $δ_{\mathrm{Ric}}>δ_c$ forces one Ricci sign globally. Conversely, for every $0<δ<δ_c$ there are local sign-changing metrics with exact pointwise pinching $δ_{\mathrm{Ric}}\equivδ$, and exact local examples also exist at $δ=δ_c$. Pointwise strictness $δ_{\mathrm{Ric}}>δ_c$ is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical $δ$ we give explicit closed metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$, and complete periodic lifts to $\mathbb{R}\times\mathbb{S}^{n-1}$, whose optimal global lower pinching constant is exactly $δ$. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.